Lucas–Lehmer primality test
In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1876^{[1]} and subsequently improved by Derrick Henry Lehmer in the 1930s.
The test[edit]
The Lucas–Lehmer test works as follows. Let M_{p} = 2^{p} − 1 be the Mersenne number to test with p an odd prime. The primality of p can be efficiently checked with a simple algorithm like trial division since p is exponentially smaller than M_{p}. Define a sequence for all i ≥ 0 by
The first few terms of this sequence are 4, 14, 194, 37634, ... (sequence A003010 in the OEIS). Then M_{p} is prime if and only if
The number s_{p − 2} mod M_{p} is called the Lucas–Lehmer residue of p. (Some authors equivalently set s_{1} = 4 and test s_{p−1} mod M_{p}). In pseudocode, the test might be written as
// Determine if M_{p} = 2^{p} − 1 is prime for p > 2 Lucas–Lehmer(p) var s = 4 var M = 2^{p} − 1 repeat p − 2 times: s = ((s × s) − 2) mod M if s == 0 return PRIME else return COMPOSITE
Performing the mod M
at each iteration ensures that all intermediate results are at most p bits (otherwise the number of bits would double each iteration). The same strategy is used in modular exponentiation.
Alternate starting values[edit]
Starting values s_{0} other than 4 are possible, for instance 10, 52, and others (sequence A018844 in the OEIS). The Lucas-Lehmer residue calculated with these alternative starting values will still be zero if M_{p} is a Mersenne prime. However, the terms of the sequence will be different and a non-zero Lucas-Lehmer residue for non-prime M_{p} will have a different numerical value from the non-zero value calculated when s_{0} = 4.
It is also possible to use the starting value (2 mod M_{p})(3 mod M_{p})^{−1}, usually denoted by 2/3 for short.^{[2]} This starting value equals (2^{p} + 1) /3, the Wagstaff number with exponent p.
Starting values like 4, 10, and 2/3 are universal, that is, they are valid for all (or nearly all) p. There are infinitely many additional universal starting values.^{[2]} However, some other starting values are only valid for a subset of all possible p, for example s_{0} = 3 can be used if p = 3 (mod 4).^{[3]} This starting value was often used where suitable in the era of hand computation, including by Lucas in proving M_{127} prime.^{[4]} The first few terms of the sequence are 3, 7, 47, ... (sequence A001566 in the OEIS).
Sign of penultimate term[edit]
If s_{p−2} = 0 mod M_{p} then the penultimate term is s_{p−3} = ± 2^{(p+1)/2} mod M_{p}. The sign of this penultimate term is called the Lehmer symbol ϵ(s_{0}, p).
In 2000 S.Y. Gebre-Egziabher proved that for the starting value 2/3 and for p ≠ 5 the sign is:
That is, ϵ(2/3, p) = +1 iff p = 1 (mod 4) and p ≠ 5.^{[2]}
The same author also proved that the Lehmer symbols for starting values 4 and 10 when p is not 2 or 5 are related by:
That is, ϵ(4, p) × ϵ(10, p) = 1 iff p = 5 or 7 (mod 8) and p ≠ 2, 5.^{[2]}
OEIS sequence A123271 shows ϵ(4, p) for each Mersenne prime M_{p}.
Time complexity[edit]
In the algorithm as written above, there are two expensive operations during each iteration: the multiplication s × s
, and the mod M
operation. The mod M
operation can be made particularly efficient on standard binary computers by observing that
This says that the least significant n bits of k plus the remaining bits of k are equivalent to k modulo 2^{n}−1. This equivalence can be used repeatedly until at most n bits remain. In this way, the remainder after dividing k by the Mersenne number 2^{n}−1 is computed without using division. For example,
916 mod 2^{5}−1 | = | 1110010100_{2} mod 2^{5}−1 |
= | ((916 mod 2^{5}) + int(916 ÷ 2^{5})) mod 2^{5}−1 | |
= | (10100_{2} + 11100_{2}) mod 2^{5}−1 | |
= | 110000_{2} mod 2^{5}−1 | |
= | (10000_{2} + 1_{2}) mod 2^{5}−1 | |
= | 10001_{2} mod 2^{5}−1 | |
= | 10001_{2} | |
= | 17. |
Moreover, since s × s
will never exceed M^{2} < 2^{2p}, this simple technique converges in at most 1 p-bit addition (and possibly a carry from the pth bit to the 1st bit), which can be done in linear time. This algorithm has a small exceptional case. It will produce 2^{n}−1 for a multiple of the modulus rather than the correct value of 0. However, this case is easy to detect and correct.
With the modulus out of the way, the asymptotic complexity of the algorithm only depends on the multiplication algorithm used to square s at each step. The simple "grade-school" algorithm for multiplication requires O(p^{2}) bit-level or word-level operations to square a p-bit number. Since this happens O(p) times, the total time complexity is O(p^{3}). A more efficient multiplication algorithm is the Schönhage–Strassen algorithm, which is based on the Fast Fourier transform. It only requires O(p log p log log p) time to square a p-bit number. This reduces the complexity to O(p^{2} log p log log p) or Õ(p^{2}).^{[5]} An even more efficient multiplication algorithm, Fürer's algorithm, only needs time to multiply two p-bit numbers.
By comparison, the most efficient randomized primality test for general integers, the Miller–Rabin primality test, requires O(k n^{2} log n log log n) bit operations using FFT multiplication for an n-digit number, where k is the number of iterations and is related to the error rate. For constant k, this is in the same complexity class as the Lucas-Lehmer test. In practice however, the cost of doing many iterations and other differences leads to worse performance for Miller–Rabin. The most efficient deterministic primality test for any n-digit number, the AKS primality test, requires Õ(n^{6}) bit operations in its best known variant and is extremely slow even for relatively small values.
Examples[edit]
The Mersenne number M_{3} = 2^{3}−1 = 7 is prime. The Lucas–Lehmer test verifies this as follows. Initially s is set to 4 and then is updated 3−2 = 1 time:
- s ← ((4 × 4) − 2) mod 7 = 0.
Since the final value of s is 0, the conclusion is that M_{3} is prime.
On the other hand, M_{11} = 2047 = 23 × 89 is not prime. Again, s is set to 4 but is now updated 11−2 = 9 times:
- s ← ((4 × 4) − 2) mod 2047 = 14
- s ← ((14 × 14) − 2) mod 2047 = 194
- s ← ((194 × 194) − 2) mod 2047 = 788
- s ← ((788 × 788) − 2) mod 2047 = 701
- s ← ((701 × 701) − 2) mod 2047 = 119
- s ← ((119 × 119) − 2) mod 2047 = 1877
- s ← ((1877 × 1877) − 2) mod 2047 = 240
- s ← ((240 × 240) − 2) mod 2047 = 282
- s ← ((282 × 282) − 2) mod 2047 = 1736
Since the final value of s is not 0, the conclusion is that M_{11} = 2047 is not prime. Although M_{11} = 2047 has nontrivial factors, the Lucas–Lehmer test gives no indication about what they might be.
Proof of correctness[edit]
The proof of correctness for this test presented here is simpler than the original proof given by Lehmer. Recall the definition
The goal is to show that M_{p} is prime iff
The sequence is a recurrence relation with a closed-form solution. Let and . It then follows by induction that for all i:
and
The last step uses This closed form is used in both the proof of sufficiency and the proof of necessity.
Sufficiency[edit]
The goal is to show that implies that is prime. What follows is a straightforward proof exploiting elementary group theory given by J. W. Bruce^{[6]} as related by Jason Wojciechowski.^{[7]}
Suppose Then
for some integer k, so
Multiplying by gives
Thus,
For a contradiction, suppose M_{p} is composite, and let q be the smallest prime factor of M_{p}. Mersenne numbers are odd, so q > 2. Informally,^{[note 1]} let be the integers modulo q, and let Multiplication in is defined as
Clearly, this multiplication is closed, i.e. the product of numbers from X is itself in X. The size of X is denoted by
Since q > 2, it follows that and are in X.^{[note 2]} The subset of elements in X with inverses forms a group, which is denoted by X* and has size One element in X that does not have an inverse is 0, so
Now and , so
in X. Then by equation (1),
in X, and squaring both sides gives
Thus lies in X* and has inverse Furthermore, the order of divides However , so the order does not divide Thus, the order is exactly
The order of an element is at most the order (size) of the group, so
But q is the smallest prime factor of the composite , so
This yields the contradiction . Therefore, is prime.
Necessity[edit]
In the other direction, the goal is to show that the primality of implies . The following simplified proof is by Öystein J. Rödseth.^{[8]}
Since for odd , it follows from properties of the Legendre symbol that This means that 3 is a quadratic nonresidue modulo By Euler's criterion, this is equivalent to
In contrast, 2 is a quadratic residue modulo since and so This time, Euler's criterion gives
Combining these two equivalence relations yields
Let , and define X as before as the ring Then in the ring X, it follows that
where the first equality uses the Binomial Theorem in a finite field, which is
- ,
and the second equality uses Fermat's little theorem, which is
for any integer a. The value of was chosen so that Consequently, this can be used to compute in the ring X as
All that remains is to multiply both sides of this equation by and use , which gives
Since is 0 in X, it is also 0 modulo
Applications[edit]
The Lucas–Lehmer test is the primality test used by the Great Internet Mersenne Prime Search (GIMPS) to locate large primes. This search has been successful in locating many of the largest primes known to date.^{[9]} The test is considered valuable because it can probably test a large set of very large numbers for primality within an affordable amount of time. In contrast, the equivalently fast Pépin's test for any Fermat number can only be used on a much smaller set of very large numbers before reaching computational limits.
See also[edit]
Notes[edit]
References[edit]
- ^ Lehmer, Derrick (1927). "Tests for primality by the converse of Fermat's theorem". Bulletin of the American Mathematical Society. American Mathematical Society. 33 (3): 327.
- ^ ^{a} ^{b} ^{c} ^{d} Jansen, B.J.H. (2012). Mersenne primes and class field theory (Doctoral thesis). Leiden University. p. iii. Retrieved 2018-12-17.
- ^ Robinson, Raphael M. (1954). "Mersenne and Fermat numbers". Proc. Amer. Math. Soc. 5: 842–846. doi:10.1090/S0002-9939-1954-0064787-4.
- ^ Haworth, Guy (1990). Mersenne numbers (PDF) (Technical report). p. 20. Retrieved 2018-12-17.
- ^ Colquitt, W. N.; Welsh, L., Jr. (1991), "A New Mersenne Prime", Mathematics of Computation, 56 (194): 867–870, doi:10.2307/2008415,
The use of the FFT speeds up the asymptotic time for the Lucas–Lehmer test for M_{p} from O(p^{3}) to O(p^{2} log p log log p) bit operations.
- ^ Bruce, J. W. (1993). "A Really Trivial Proof of the Lucas–Lehmer Test". The American Mathematical Monthly. 100 (4): 370–371. doi:10.2307/2324959.
- ^ Jason Wojciechowski. Mersenne Primes, An Introduction and Overview. 2003.
- ^ Rödseth, Öystein J. (1994). "A note on primality tests for N=h·2^n−1" (PDF). BIT Numerical Mathematics. 34 (3): 451–454. doi:10.1007/BF01935653. Archived from the original (PDF) on March 6, 2016.
- ^ The "Top Ten" Record Primes, The Prime Pages
- Crandall, Richard; Pomerance, Carl (2001), "Section 4.2.1: The Lucas–Lehmer test", Prime Numbers: A Computational Perspective (1st ed.), Berlin: Springer, p. 167–170, ISBN 0-387-94777-9